Asymptotic Properties of Multicolor Randomly Reinforced Pólya Urns

Abstract

The generalized Pólya urn has been extensively studied and is widely applied in many disciplines. An important application of urn models is in the development of randomized treatment allocation schemes in clinical studies. The randomly reinforced urn was recently proposed, but, although the model has some intuitively desirable properties, it lacks theoretical justification. In this paper we obtain important asymptotic properties for multicolor reinforced urn models. We derive results for the rate of convergence of the number of patients assigned to each treatment under a set of minimum required conditions and provide the distributions of the limits. Furthermore, we study the asymptotic behavior for the nonhomogeneous case.

Infinite-color randomly reinforced urns with dominant colors

“…, k 0 }. In [21,22] the authors look at the case k 0 = 1 and k 0 = k, whereas [4,26] consider 1 < k 0 < k, although [4] assumes (3.2) only asymptotically.…”

“…Then we expect that the colors which have a higher expected reinforcement will tend to dominate the observation process, just as better-adapted species have a greater chance at survival. To the best of our knowledge, existing studies consider only the model with k colors, which is known as a randomly reinforced urn (RRU), and show that the probability of drawing a color whose expected weight is maximum goes indeed to one, see, e.g., [4,21,22,26]. We thus explore the extension of RRUs to an unbounded set of possible colors, which parallels the extension provided by exchangeable Pólya sequences for k-color Pólya urns.…”

A Gaussian process approximation for two-color randomly reinforced urns

“…It is of interest to find the distribution of ψ ∞ . In the case that Y 0,k and U m,k , k = 1, 2, are all integers, in a recent manuscript of Zhang, et al (2010) it is proved that the distribution of ψ ∞ is absolutely continuous and is determined by Y 0,1 , Y 0,2 and the distributions of…”

“…Finally, this paper only consider the two-color urn model. In the manuscript of Zhang, et al (2010), the asymptotic properties for a multi-color reinforced urn model are studied. It is expected to approximate the urn components after being suitably normalized by a multi-dimensional Gaussian process.…”

Central limit theorems for a hypergeometric randomly reinforced urn

“…Subsequently, urn models have been widely studied by many researchers and there is a rather extensive literature on them (e.g. [2,4,9,10,15,17,18,24,33,34,38,44]): a large number of new "replacement policies" (for instance, balanced rules, tenable mechanisms and random reinforcements) and various related models (for instance, the Poisson-Dirichlet model [6] and the very recent Indian buffet model [8]) have been introduced and analyzed from different points of view and by means of different techniques (combinatorial methods, martingales, branching processes, stochastic approximations, etc.). We refer to [40], and the references therein, for a general survey on random processes with reinforcement.…”